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Unisolvence of random Kansa collocation by Thin-Plate Splines for the Poisson equation

Francesco Dell'Accio, Alvise Sommariva, Marco Vianello

Abstract

Existence of sufficient conditions for unisolvence of Kansa unsymmetric collocation for PDEs is still an open problem. In this paper we make a first step in this direction, proving that unsymmetric collocation matrices with Thin-Plate Splines for the 2D Poisson equation are almost surely nonsingular, when the discretization points are chosen randomly on domains with analytic boundary.

Unisolvence of random Kansa collocation by Thin-Plate Splines for the Poisson equation

Abstract

Existence of sufficient conditions for unisolvence of Kansa unsymmetric collocation for PDEs is still an open problem. In this paper we make a first step in this direction, proving that unsymmetric collocation matrices with Thin-Plate Splines for the 2D Poisson equation are almost surely nonsingular, when the discretization points are chosen randomly on domains with analytic boundary.
Paper Structure (3 sections, 2 theorems, 37 equations)

This paper contains 3 sections, 2 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Unisolvence of random Kansa collocation
  3. Remarks on possible extensions

Key Result

Theorem 1

Let $K_N$ be the TPS Kansa collocation matrix defined above, with $N=n+m\geq 2$, where $\{P_i\}$ is a sequence of independent uniformly distributed random points in $\Omega$, and $\{Q_h\}$ a sequence of independent uniformly distributed points on $\partial\Omega$. Namely, $\{Q_h\}=\{\gamma(t_h)\}$ w

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2